The only service I know the City supplies to the Town is that some (not all) Town residents do get city water. The agreement is decades old and difference in rates is being done away with. Seneca Falls and Lyons were Villages and Geneva is a City, it is much more complicated to dissolve a City.
HWS was worried about paying a "street benefit". They own property on South Main, Pultney, Hamilton and St. Clare and that is a lot of frontage. But it was not only HWS, the Station, Geneva General, and every not-for-profit was involved and they did not want this charge either. The pastors of both the Roman Catholic and Presbyterian churches made appearances in front of Council in opposition and others contacted their own Council members. Some of them actually said that they would have to withdraw from community programs or even close their doors if this went through.
There were various rates discussed. One for single family owner occupied, one for multiple family residences, corner properties getting a "discount" on one of the street fronts, and a special rate for other not-for-profits. And what about HWS and their corner properties? They had to be included or the district would be illegal.
All of this became too complicated to be workable.
Some property owners would have paid more under this plan. This is not theoretical as specific properties were actually identified.
Here is an example I knew of: an elderly couple living in a multiple family residence on a corner lot, one side of which was a double lot. They did not have the rental unit rented, but they were still listed as multiple family on the tax rolls. Because of their age, military service, etc. they were taxed at less than assessed value, but they would have gone away with the benefit district.
Some members of the Council said they would support this benefit district only if every one's taxes decreased and that was not going to happen, but in the end it was the complexity of trying to figure out who what pay which rates that defeated this plan.
I hope this answers some of the questions.